PulseFocusPlatform/ppdet/data/transform/mot_operators.py

635 lines
25 KiB
Python

# Copyright (c) 2021 PaddlePaddle Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
try:
from collections.abc import Sequence
except Exception:
from collections import Sequence
from numbers import Integral
import cv2
import copy
import numpy as np
import random
import math
from .operators import BaseOperator, register_op
from .batch_operators import Gt2TTFTarget
from ppdet.modeling.bbox_utils import bbox_iou_np_expand
from ppdet.utils.logger import setup_logger
logger = setup_logger(__name__)
__all__ = [
'RGBReverse', 'LetterBoxResize', 'MOTRandomAffine', 'Gt2JDETargetThres',
'Gt2JDETargetMax', 'Gt2FairMOTTarget'
]
@register_op
class RGBReverse(BaseOperator):
"""RGB to BGR, or BGR to RGB, sensitive to MOTRandomAffine
"""
def __init__(self):
super(RGBReverse, self).__init__()
def apply(self, sample, context=None):
im = sample['image']
sample['image'] = np.ascontiguousarray(im[:, :, ::-1])
return sample
@register_op
class LetterBoxResize(BaseOperator):
def __init__(self, target_size):
"""
Resize image to target size, convert normalized xywh to pixel xyxy
format ([x_center, y_center, width, height] -> [x0, y0, x1, y1]).
Args:
target_size (int|list): image target size.
"""
super(LetterBoxResize, self).__init__()
if not isinstance(target_size, (Integral, Sequence)):
raise TypeError(
"Type of target_size is invalid. Must be Integer or List or Tuple, now is {}".
format(type(target_size)))
if isinstance(target_size, Integral):
target_size = [target_size, target_size]
self.target_size = target_size
def apply_image(self, img, height, width, color=(127.5, 127.5, 127.5)):
# letterbox: resize a rectangular image to a padded rectangular
shape = img.shape[:2] # [height, width]
ratio_h = float(height) / shape[0]
ratio_w = float(width) / shape[1]
ratio = min(ratio_h, ratio_w)
new_shape = (round(shape[1] * ratio),
round(shape[0] * ratio)) # [width, height]
padw = (width - new_shape[0]) / 2
padh = (height - new_shape[1]) / 2
top, bottom = round(padh - 0.1), round(padh + 0.1)
left, right = round(padw - 0.1), round(padw + 0.1)
img = cv2.resize(
img, new_shape, interpolation=cv2.INTER_AREA) # resized, no border
img = cv2.copyMakeBorder(
img, top, bottom, left, right, cv2.BORDER_CONSTANT,
value=color) # padded rectangular
return img, ratio, padw, padh
def apply_bbox(self, bbox0, h, w, ratio, padw, padh):
bboxes = bbox0.copy()
bboxes[:, 0] = ratio * w * (bbox0[:, 0] - bbox0[:, 2] / 2) + padw
bboxes[:, 1] = ratio * h * (bbox0[:, 1] - bbox0[:, 3] / 2) + padh
bboxes[:, 2] = ratio * w * (bbox0[:, 0] + bbox0[:, 2] / 2) + padw
bboxes[:, 3] = ratio * h * (bbox0[:, 1] + bbox0[:, 3] / 2) + padh
return bboxes
def apply(self, sample, context=None):
""" Resize the image numpy.
"""
im = sample['image']
h, w = sample['im_shape']
if not isinstance(im, np.ndarray):
raise TypeError("{}: image type is not numpy.".format(self))
if len(im.shape) != 3:
raise ImageError('{}: image is not 3-dimensional.'.format(self))
# apply image
height, width = self.target_size
img, ratio, padw, padh = self.apply_image(
im, height=height, width=width)
sample['image'] = img
new_shape = (round(h * ratio), round(w * ratio))
sample['im_shape'] = np.asarray(new_shape, dtype=np.float32)
sample['scale_factor'] = np.asarray([ratio, ratio], dtype=np.float32)
# apply bbox
if 'gt_bbox' in sample and len(sample['gt_bbox']) > 0:
sample['gt_bbox'] = self.apply_bbox(sample['gt_bbox'], h, w, ratio,
padw, padh)
return sample
@register_op
class MOTRandomAffine(BaseOperator):
"""
Affine transform to image and coords to achieve the rotate, scale and
shift effect for training image.
Args:
degrees (list[2]): the rotate range to apply, transform range is [min, max]
translate (list[2]): the translate range to apply, ransform range is [min, max]
scale (list[2]): the scale range to apply, transform range is [min, max]
shear (list[2]): the shear range to apply, transform range is [min, max]
borderValue (list[3]): value used in case of a constant border when appling
the perspective transformation
reject_outside (bool): reject warped bounding bboxes outside of image
Returns:
records(dict): contain the image and coords after tranformed
"""
def __init__(self,
degrees=(-5, 5),
translate=(0.10, 0.10),
scale=(0.50, 1.20),
shear=(-2, 2),
borderValue=(127.5, 127.5, 127.5),
reject_outside=True):
super(MOTRandomAffine, self).__init__()
self.degrees = degrees
self.translate = translate
self.scale = scale
self.shear = shear
self.borderValue = borderValue
self.reject_outside = reject_outside
def apply(self, sample, context=None):
# https://medium.com/uruvideo/dataset-augmentation-with-random-homographies-a8f4b44830d4
border = 0 # width of added border (optional)
img = sample['image']
height, width = img.shape[0], img.shape[1]
# Rotation and Scale
R = np.eye(3)
a = random.random() * (self.degrees[1] - self.degrees[0]
) + self.degrees[0]
s = random.random() * (self.scale[1] - self.scale[0]) + self.scale[0]
R[:2] = cv2.getRotationMatrix2D(
angle=a, center=(width / 2, height / 2), scale=s)
# Translation
T = np.eye(3)
T[0, 2] = (
random.random() * 2 - 1
) * self.translate[0] * height + border # x translation (pixels)
T[1, 2] = (
random.random() * 2 - 1
) * self.translate[1] * width + border # y translation (pixels)
# Shear
S = np.eye(3)
S[0, 1] = math.tan((random.random() *
(self.shear[1] - self.shear[0]) + self.shear[0]) *
math.pi / 180) # x shear (deg)
S[1, 0] = math.tan((random.random() *
(self.shear[1] - self.shear[0]) + self.shear[0]) *
math.pi / 180) # y shear (deg)
M = S @T @R # Combined rotation matrix. ORDER IS IMPORTANT HERE!!
imw = cv2.warpPerspective(
img,
M,
dsize=(width, height),
flags=cv2.INTER_LINEAR,
borderValue=self.borderValue) # BGR order borderValue
if 'gt_bbox' in sample and len(sample['gt_bbox']) > 0:
targets = sample['gt_bbox']
n = targets.shape[0]
points = targets.copy()
area0 = (points[:, 2] - points[:, 0]) * (
points[:, 3] - points[:, 1])
# warp points
xy = np.ones((n * 4, 3))
xy[:, :2] = points[:, [0, 1, 2, 3, 0, 3, 2, 1]].reshape(
n * 4, 2) # x1y1, x2y2, x1y2, x2y1
xy = (xy @M.T)[:, :2].reshape(n, 8)
# create new boxes
x = xy[:, [0, 2, 4, 6]]
y = xy[:, [1, 3, 5, 7]]
xy = np.concatenate(
(x.min(1), y.min(1), x.max(1), y.max(1))).reshape(4, n).T
# apply angle-based reduction
radians = a * math.pi / 180
reduction = max(abs(math.sin(radians)), abs(math.cos(radians)))**0.5
x = (xy[:, 2] + xy[:, 0]) / 2
y = (xy[:, 3] + xy[:, 1]) / 2
w = (xy[:, 2] - xy[:, 0]) * reduction
h = (xy[:, 3] - xy[:, 1]) * reduction
xy = np.concatenate(
(x - w / 2, y - h / 2, x + w / 2, y + h / 2)).reshape(4, n).T
# reject warped points outside of image
if self.reject_outside:
np.clip(xy[:, 0], 0, width, out=xy[:, 0])
np.clip(xy[:, 2], 0, width, out=xy[:, 2])
np.clip(xy[:, 1], 0, height, out=xy[:, 1])
np.clip(xy[:, 3], 0, height, out=xy[:, 3])
w = xy[:, 2] - xy[:, 0]
h = xy[:, 3] - xy[:, 1]
area = w * h
ar = np.maximum(w / (h + 1e-16), h / (w + 1e-16))
i = (w > 4) & (h > 4) & (area / (area0 + 1e-16) > 0.1) & (ar < 10)
if sum(i) > 0:
sample['gt_bbox'] = xy[i].astype(sample['gt_bbox'].dtype)
sample['gt_class'] = sample['gt_class'][i]
if 'difficult' in sample:
sample['difficult'] = sample['difficult'][i]
if 'gt_ide' in sample:
sample['gt_ide'] = sample['gt_ide'][i]
if 'is_crowd' in sample:
sample['is_crowd'] = sample['is_crowd'][i]
sample['image'] = imw
return sample
else:
return sample
@register_op
class Gt2JDETargetThres(BaseOperator):
__shared__ = ['num_classes']
"""
Generate JDE targets by groud truth data when training
Args:
anchors (list): anchors of JDE model
anchor_masks (list): anchor_masks of JDE model
downsample_ratios (list): downsample ratios of JDE model
ide_thresh (float): thresh of identity, higher is groud truth
fg_thresh (float): thresh of foreground, higher is foreground
bg_thresh (float): thresh of background, lower is background
num_classes (int): number of classes
"""
def __init__(self,
anchors,
anchor_masks,
downsample_ratios,
ide_thresh=0.5,
fg_thresh=0.5,
bg_thresh=0.4,
num_classes=1):
super(Gt2JDETargetThres, self).__init__()
self.anchors = anchors
self.anchor_masks = anchor_masks
self.downsample_ratios = downsample_ratios
self.ide_thresh = ide_thresh
self.fg_thresh = fg_thresh
self.bg_thresh = bg_thresh
self.num_classes = num_classes
def generate_anchor(self, nGh, nGw, anchor_hw):
nA = len(anchor_hw)
yy, xx = np.meshgrid(np.arange(nGh), np.arange(nGw))
mesh = np.stack([xx.T, yy.T], axis=0) # [2, nGh, nGw]
mesh = np.repeat(mesh[None, :], nA, axis=0) # [nA, 2, nGh, nGw]
anchor_offset_mesh = anchor_hw[:, :, None][:, :, :, None]
anchor_offset_mesh = np.repeat(anchor_offset_mesh, nGh, axis=-2)
anchor_offset_mesh = np.repeat(anchor_offset_mesh, nGw, axis=-1)
anchor_mesh = np.concatenate(
[mesh, anchor_offset_mesh], axis=1) # [nA, 4, nGh, nGw]
return anchor_mesh
def encode_delta(self, gt_box_list, fg_anchor_list):
px, py, pw, ph = fg_anchor_list[:, 0], fg_anchor_list[:,1], \
fg_anchor_list[:, 2], fg_anchor_list[:,3]
gx, gy, gw, gh = gt_box_list[:, 0], gt_box_list[:, 1], \
gt_box_list[:, 2], gt_box_list[:, 3]
dx = (gx - px) / pw
dy = (gy - py) / ph
dw = np.log(gw / pw)
dh = np.log(gh / ph)
return np.stack([dx, dy, dw, dh], axis=1)
def pad_box(self, sample, num_max):
assert 'gt_bbox' in sample
bbox = sample['gt_bbox']
gt_num = len(bbox)
pad_bbox = np.zeros((num_max, 4), dtype=np.float32)
if gt_num > 0:
pad_bbox[:gt_num, :] = bbox[:gt_num, :]
sample['gt_bbox'] = pad_bbox
if 'gt_score' in sample:
pad_score = np.zeros((num_max, ), dtype=np.float32)
if gt_num > 0:
pad_score[:gt_num] = sample['gt_score'][:gt_num, 0]
sample['gt_score'] = pad_score
if 'difficult' in sample:
pad_diff = np.zeros((num_max, ), dtype=np.int32)
if gt_num > 0:
pad_diff[:gt_num] = sample['difficult'][:gt_num, 0]
sample['difficult'] = pad_diff
if 'is_crowd' in sample:
pad_crowd = np.zeros((num_max, ), dtype=np.int32)
if gt_num > 0:
pad_crowd[:gt_num] = sample['is_crowd'][:gt_num, 0]
sample['is_crowd'] = pad_crowd
if 'gt_ide' in sample:
pad_ide = np.zeros((num_max, ), dtype=np.int32)
if gt_num > 0:
pad_ide[:gt_num] = sample['gt_ide'][:gt_num, 0]
sample['gt_ide'] = pad_ide
return sample
def __call__(self, samples, context=None):
assert len(self.anchor_masks) == len(self.downsample_ratios), \
"anchor_masks', and 'downsample_ratios' should have same length."
h, w = samples[0]['image'].shape[1:3]
num_max = 0
for sample in samples:
num_max = max(num_max, len(sample['gt_bbox']))
for sample in samples:
gt_bbox = sample['gt_bbox']
gt_ide = sample['gt_ide']
for i, (anchor_hw, downsample_ratio
) in enumerate(zip(self.anchors, self.downsample_ratios)):
anchor_hw = np.array(
anchor_hw, dtype=np.float32) / downsample_ratio
nA = len(anchor_hw)
nGh, nGw = int(h / downsample_ratio), int(w / downsample_ratio)
tbox = np.zeros((nA, nGh, nGw, 4), dtype=np.float32)
tconf = np.zeros((nA, nGh, nGw), dtype=np.float32)
tid = -np.ones((nA, nGh, nGw, 1), dtype=np.float32)
gxy, gwh = gt_bbox[:, 0:2].copy(), gt_bbox[:, 2:4].copy()
gxy[:, 0] = gxy[:, 0] * nGw
gxy[:, 1] = gxy[:, 1] * nGh
gwh[:, 0] = gwh[:, 0] * nGw
gwh[:, 1] = gwh[:, 1] * nGh
gxy[:, 0] = np.clip(gxy[:, 0], 0, nGw - 1)
gxy[:, 1] = np.clip(gxy[:, 1], 0, nGh - 1)
tboxes = np.concatenate([gxy, gwh], axis=1)
anchor_mesh = self.generate_anchor(nGh, nGw, anchor_hw)
anchor_list = np.transpose(anchor_mesh,
(0, 2, 3, 1)).reshape(-1, 4)
iou_pdist = bbox_iou_np_expand(
anchor_list, tboxes, x1y1x2y2=False)
iou_max = np.max(iou_pdist, axis=1)
max_gt_index = np.argmax(iou_pdist, axis=1)
iou_map = iou_max.reshape(nA, nGh, nGw)
gt_index_map = max_gt_index.reshape(nA, nGh, nGw)
id_index = iou_map > self.ide_thresh
fg_index = iou_map > self.fg_thresh
bg_index = iou_map < self.bg_thresh
ign_index = (iou_map < self.fg_thresh) * (
iou_map > self.bg_thresh)
tconf[fg_index] = 1
tconf[bg_index] = 0
tconf[ign_index] = -1
gt_index = gt_index_map[fg_index]
gt_box_list = tboxes[gt_index]
gt_id_list = gt_ide[gt_index_map[id_index]]
if np.sum(fg_index) > 0:
tid[id_index] = gt_id_list
fg_anchor_list = anchor_list.reshape(nA, nGh, nGw,
4)[fg_index]
delta_target = self.encode_delta(gt_box_list,
fg_anchor_list)
tbox[fg_index] = delta_target
sample['tbox{}'.format(i)] = tbox
sample['tconf{}'.format(i)] = tconf
sample['tide{}'.format(i)] = tid
sample.pop('gt_class')
sample = self.pad_box(sample, num_max)
return samples
@register_op
class Gt2JDETargetMax(BaseOperator):
__shared__ = ['num_classes']
"""
Generate JDE targets by groud truth data when evaluating
Args:
anchors (list): anchors of JDE model
anchor_masks (list): anchor_masks of JDE model
downsample_ratios (list): downsample ratios of JDE model
max_iou_thresh (float): iou thresh for high quality anchor
num_classes (int): number of classes
"""
def __init__(self,
anchors,
anchor_masks,
downsample_ratios,
max_iou_thresh=0.60,
num_classes=1):
super(Gt2JDETargetMax, self).__init__()
self.anchors = anchors
self.anchor_masks = anchor_masks
self.downsample_ratios = downsample_ratios
self.max_iou_thresh = max_iou_thresh
self.num_classes = num_classes
def __call__(self, samples, context=None):
assert len(self.anchor_masks) == len(self.downsample_ratios), \
"anchor_masks', and 'downsample_ratios' should have same length."
h, w = samples[0]['image'].shape[1:3]
for sample in samples:
gt_bbox = sample['gt_bbox']
gt_ide = sample['gt_ide']
for i, (anchor_hw, downsample_ratio
) in enumerate(zip(self.anchors, self.downsample_ratios)):
anchor_hw = np.array(
anchor_hw, dtype=np.float32) / downsample_ratio
nA = len(anchor_hw)
nGh, nGw = int(h / downsample_ratio), int(w / downsample_ratio)
tbox = np.zeros((nA, nGh, nGw, 4), dtype=np.float32)
tconf = np.zeros((nA, nGh, nGw), dtype=np.float32)
tid = -np.ones((nA, nGh, nGw, 1), dtype=np.float32)
gxy, gwh = gt_bbox[:, 0:2].copy(), gt_bbox[:, 2:4].copy()
gxy[:, 0] = gxy[:, 0] * nGw
gxy[:, 1] = gxy[:, 1] * nGh
gwh[:, 0] = gwh[:, 0] * nGw
gwh[:, 1] = gwh[:, 1] * nGh
gi = np.clip(gxy[:, 0], 0, nGw - 1).astype(int)
gj = np.clip(gxy[:, 1], 0, nGh - 1).astype(int)
# iou of targets-anchors (using wh only)
box1 = gwh
box2 = anchor_hw[:, None, :]
inter_area = np.minimum(box1, box2).prod(2)
iou = inter_area / (
box1.prod(1) + box2.prod(2) - inter_area + 1e-16)
# Select best iou_pred and anchor
iou_best = iou.max(0) # best anchor [0-2] for each target
a = np.argmax(iou, axis=0)
# Select best unique target-anchor combinations
iou_order = np.argsort(-iou_best) # best to worst
# Unique anchor selection
u = np.stack((gi, gj, a), 0)[:, iou_order]
_, first_unique = np.unique(u, axis=1, return_index=True)
mask = iou_order[first_unique]
# best anchor must share significant commonality (iou) with target
# TODO: examine arbitrary threshold
idx = mask[iou_best[mask] > self.max_iou_thresh]
if len(idx) > 0:
a_i, gj_i, gi_i = a[idx], gj[idx], gi[idx]
t_box = gt_bbox[idx]
t_id = gt_ide[idx]
if len(t_box.shape) == 1:
t_box = t_box.reshape(1, 4)
gxy, gwh = t_box[:, 0:2].copy(), t_box[:, 2:4].copy()
gxy[:, 0] = gxy[:, 0] * nGw
gxy[:, 1] = gxy[:, 1] * nGh
gwh[:, 0] = gwh[:, 0] * nGw
gwh[:, 1] = gwh[:, 1] * nGh
# XY coordinates
tbox[:, :, :, 0:2][a_i, gj_i, gi_i] = gxy - gxy.astype(int)
# Width and height in yolo method
tbox[:, :, :, 2:4][a_i, gj_i, gi_i] = np.log(gwh /
anchor_hw[a_i])
tconf[a_i, gj_i, gi_i] = 1
tid[a_i, gj_i, gi_i] = t_id
sample['tbox{}'.format(i)] = tbox
sample['tconf{}'.format(i)] = tconf
sample['tide{}'.format(i)] = tid
class Gt2FairMOTTarget(Gt2TTFTarget):
__shared__ = ['num_classes']
"""
Generate FairMOT targets by ground truth data.
Difference between Gt2FairMOTTarget and Gt2TTFTarget are:
1. the gaussian kernal radius to generate a heatmap.
2. the targets needed during traing.
Args:
num_classes(int): the number of classes.
down_ratio(int): the down ratio from images to heatmap, 4 by default.
max_objs(int): the maximum number of ground truth objects in a image, 500 by default.
"""
def __init__(self, num_classes=1, down_ratio=4, max_objs=500):
super(Gt2TTFTarget, self).__init__()
self.down_ratio = down_ratio
self.num_classes = num_classes
self.max_objs = max_objs
def __call__(self, samples, context=None):
for b_id, sample in enumerate(samples):
output_h = sample['image'].shape[1] // self.down_ratio
output_w = sample['image'].shape[2] // self.down_ratio
heatmap = np.zeros(
(self.num_classes, output_h, output_w), dtype='float32')
bbox_size = np.zeros((self.max_objs, 4), dtype=np.float32)
center_offset = np.zeros((self.max_objs, 2), dtype=np.float32)
index = np.zeros((self.max_objs, ), dtype=np.int64)
index_mask = np.zeros((self.max_objs, ), dtype=np.int32)
reid = np.zeros((self.max_objs, ), dtype=np.int64)
bbox_xys = np.zeros((self.max_objs, 4), dtype=np.float32)
gt_bbox = sample['gt_bbox']
gt_class = sample['gt_class']
gt_ide = sample['gt_ide']
for k in range(len(gt_bbox)):
cls_id = gt_class[k][0]
bbox = gt_bbox[k]
ide = gt_ide[k][0]
bbox[[0, 2]] = bbox[[0, 2]] * output_w
bbox[[1, 3]] = bbox[[1, 3]] * output_h
bbox_amodal = copy.deepcopy(bbox)
bbox_amodal[0] = bbox_amodal[0] - bbox_amodal[2] / 2.
bbox_amodal[1] = bbox_amodal[1] - bbox_amodal[3] / 2.
bbox_amodal[2] = bbox_amodal[0] + bbox_amodal[2]
bbox_amodal[3] = bbox_amodal[1] + bbox_amodal[3]
bbox[0] = np.clip(bbox[0], 0, output_w - 1)
bbox[1] = np.clip(bbox[1], 0, output_h - 1)
h = bbox[3]
w = bbox[2]
bbox_xy = copy.deepcopy(bbox)
bbox_xy[0] = bbox_xy[0] - bbox_xy[2] / 2
bbox_xy[1] = bbox_xy[1] - bbox_xy[3] / 2
bbox_xy[2] = bbox_xy[0] + bbox_xy[2]
bbox_xy[3] = bbox_xy[1] + bbox_xy[3]
if h > 0 and w > 0:
radius = self.gaussian_radius((math.ceil(h), math.ceil(w)))
radius = max(0, int(radius))
ct = np.array([bbox[0], bbox[1]], dtype=np.float32)
ct_int = ct.astype(np.int32)
self.draw_truncate_gaussian(heatmap[cls_id], ct_int, radius,
radius)
bbox_size[k] = ct[0] - bbox_amodal[0], ct[1] - bbox_amodal[1], \
bbox_amodal[2] - ct[0], bbox_amodal[3] - ct[1]
index[k] = ct_int[1] * output_w + ct_int[0]
center_offset[k] = ct - ct_int
index_mask[k] = 1
reid[k] = ide
bbox_xys[k] = bbox_xy
sample['heatmap'] = heatmap
sample['index'] = index
sample['offset'] = center_offset
sample['size'] = bbox_size
sample['index_mask'] = index_mask
sample['reid'] = reid
sample['bbox_xys'] = bbox_xys
sample.pop('is_crowd', None)
sample.pop('difficult', None)
sample.pop('gt_class', None)
sample.pop('gt_bbox', None)
sample.pop('gt_score', None)
sample.pop('gt_ide', None)
return samples
def gaussian_radius(self, det_size, min_overlap=0.7):
height, width = det_size
a1 = 1
b1 = (height + width)
c1 = width * height * (1 - min_overlap) / (1 + min_overlap)
sq1 = np.sqrt(b1**2 - 4 * a1 * c1)
r1 = (b1 + sq1) / 2
a2 = 4
b2 = 2 * (height + width)
c2 = (1 - min_overlap) * width * height
sq2 = np.sqrt(b2**2 - 4 * a2 * c2)
r2 = (b2 + sq2) / 2
a3 = 4 * min_overlap
b3 = -2 * min_overlap * (height + width)
c3 = (min_overlap - 1) * width * height
sq3 = np.sqrt(b3**2 - 4 * a3 * c3)
r3 = (b3 + sq3) / 2
return min(r1, r2, r3)